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Critical exponents of magnets with lengthy defects
Volume 76A, number 2
PHYSICS LETTERS
17 March 1980
CRITICAL EXPONENTS OF MAGNETS WITH LENGTHY DEFECTS S.N. DOROGOVTSEV The Academy of Sciences of the USSR, A.F. loffe Physico-Technical Institute, Leningrad, 194021, USSR
Received 23 October 1979
The critical behaviour of magnets with non-zero-dimensional defects is investigated by the renormalization group method. Expansions of the critical exponents in the small parameters e and e~are obtained, where Ed is the defect dimensionality. The corresponding renormalization group equations are shown to possess a focus-type fixed point.
1. The critical exponents for defect magnets were calculated originally by Khmelnitskii [1] and Harris and Lubensky [2,3]. These authors actually studied point impurities. For the case of linear and plane defects Levanyuk et al. [4] calculated only the first corrections to the mean field theory. In the present paper the fluctuation region in magnets with lengthy defects is studied.
number of defects per unit cross-sectional area), a is the lattice spacing, ~d is the defect dimensionality, denotes the momentum components in the lengthy defect subspace,j = 1, ed. In the final formulas one should put ~d = 1. Formula (2) contains 5-functions. So in some diagrams the integrals f d4—ek are changed to fd4_E_edk. The new integrals diverge stronger than the others.
2. Let us consider a magnet with randomly distributed parallel linear defects. Each of them consists of quenched point defects arranged with the periodicity of the crystal lattice. The contribution to the hamil-
3. Let us assume that n > 1. In lowest order the Gell-Mann---Low equations are of the following form:
tonian from each point impurity is
dii’Idx
...
=
I
—
n’),
Id~crV
2
(2inIz2)~ 1 6(A~,),
(2)
—
v[
d’v/dx
2e
2~(r ~ = 1,2,~.,n~ (l) H~= Vf drI~(r)I where Vis the impurity potential strength, ~‘~} is an n-component order parameter. The remaining part of the hamiltonian is of the conventional form for the X~p4theory. There are correlations between the point impurities within a lengthy defect. So we can average diagrams over different arrangements of the linear defects directly. Now the broken line in the conventional finpurity diagram technique [5] is given by
,
=
~(n + 8)y ÷6~fl — ~ (n + 2)T+
il’[~ (e + ~d)
411],
where Tare the [1] invariant corresponding to the l~and random vertex and tocharges the usual one,x = log ~2, ~is the correlation length. For the case Ed = 0 we have the point defect situation [1—3].Fig. 1 shows the phase plane for the system (3) (fig. la corresponds to e >0, Cd >0, fig. lb to = 0, ~d >0). The initial values of {T~l?’~are situated in the first quadrant of the phase plane near the coordinate origin. The random fixed point {Td, Ud} is a focus: 7d
=
ud
=
~ (~ + 3 ~d)I(~ — 1), ~ [(4— n)e + (n + 8)ed]/(n
(4) —
1).
The eigenvalues of eqs. (3) linearized in the vicinity of{Td,l~d}are
where Ccr is the concentration of lengthy defects (the 169
Volume 76A, number 2
PHYSICS LETTERS
a
\\\
4. When n = 1 the system (3) is accidentally degenerate. So it is necessary to take into account the next order of the Gell-Mann—Low functions and to carry
_________
Fig. 1. Phase
~
_________
plane for eqs.(3). (a) e >
0, Cd> 0; (b)
=
0,
Ed> 0. X ±j~= [16(n 1)]1[A A = —[3ne + (n + 8)~d]
±i(4B
—
B = 4(e When
+ 36d)(n
—
—
A 2)1/2] (5)
1)[(4
n) + (n + 8)d]
—
of~ie
When e = 0 we get the critical exponents for uniaxial magnets with lengthy defects. They are different from the mean field exponents. In conclusion, the author would like to note that a focus in the phase plane of the Gell-Mann—Low equations may be found in other systems too. Khmelnitskii [6] showed that the renormalization group equations for the phase transition at T = 0 in defect magnets possesses a stable focus fixed point. Korzhenevskii [8] found a focus-type fixed point while investigating the Potts model.
.
0
=
17 March 1980
(6) It is known that the presence of a stable focus in the phase plane leads to oscillations in the corrections to the scaling laws [6]. The susceptibility, for instance, is given by
The author is grateful to Yu A. Firsov, S.A. Ktitorov and B.N. Shalaev for useful discussions. References Khmelnitskii, Zh. Eksp. Teor. Fiz. 68(1975)1960. A.B. Harris and T.C. Lubensky, Phys. Rev. Lett. 33 (1974)
[1] D.E.
x~’
—
r1 exp [Ci-Xcos(v log r + p)],
(7)
where r = (T Tc)/Tc, T> T~C, p are non-universal constants. It can be shown that the , 6d expansions of the critical exponents are given by —
7
=
1
+
3
n
+
1 Sn
i~,~
4
+
—
~
+
“.‘
(8) a = [8(n
170
—
1)]~[(n —4)
—
(Sn +4)d]
+
...
.
[2]
1540. [3] T.C. Lubensky, Phys. Rev. Bil (1975) 3573. [4] A.P. Levanyuk, V.V. Osipov, A.S. Sigov and A.A.
Sobyanin, Zh. Eksp. Teor. Fiz.
76 (1979) 345.
[5] A.A. Abrikosov, L.P. Gor’kov and I.E. Dzyaloshinskii, Quantum field theory
methods in statistical physics (Benjamin, New York, 1963). [6] D.E. Khmelnitskii, Phys. Left. 67A (1978) 59. [7] B.N. Shalaev, Zh. Eksp. Teor. Fiz. 73 (1977) 2301. [8] A.L. Korzhenevskü, Zh. Eksp. Teor. Fiz. 75(1978)1474.
PHYSICS LETTERS
17 March 1980
CRITICAL EXPONENTS OF MAGNETS WITH LENGTHY DEFECTS S.N. DOROGOVTSEV The Academy of Sciences of the USSR, A.F. loffe Physico-Technical Institute, Leningrad, 194021, USSR
Received 23 October 1979
The critical behaviour of magnets with non-zero-dimensional defects is investigated by the renormalization group method. Expansions of the critical exponents in the small parameters e and e~are obtained, where Ed is the defect dimensionality. The corresponding renormalization group equations are shown to possess a focus-type fixed point.
1. The critical exponents for defect magnets were calculated originally by Khmelnitskii [1] and Harris and Lubensky [2,3]. These authors actually studied point impurities. For the case of linear and plane defects Levanyuk et al. [4] calculated only the first corrections to the mean field theory. In the present paper the fluctuation region in magnets with lengthy defects is studied.
number of defects per unit cross-sectional area), a is the lattice spacing, ~d is the defect dimensionality, denotes the momentum components in the lengthy defect subspace,j = 1, ed. In the final formulas one should put ~d = 1. Formula (2) contains 5-functions. So in some diagrams the integrals f d4—ek are changed to fd4_E_edk. The new integrals diverge stronger than the others.
2. Let us consider a magnet with randomly distributed parallel linear defects. Each of them consists of quenched point defects arranged with the periodicity of the crystal lattice. The contribution to the hamil-
3. Let us assume that n > 1. In lowest order the Gell-Mann---Low equations are of the following form:
tonian from each point impurity is
dii’Idx
...
=
I
—
n’),
Id~crV
2
(2inIz2)~ 1 6(A~,),
(2)
—
v[
d’v/dx
2e
2~(r ~ = 1,2,~.,n~ (l) H~= Vf drI~(r)I where Vis the impurity potential strength, ~‘~} is an n-component order parameter. The remaining part of the hamiltonian is of the conventional form for the X~p4theory. There are correlations between the point impurities within a lengthy defect. So we can average diagrams over different arrangements of the linear defects directly. Now the broken line in the conventional finpurity diagram technique [5] is given by
,
=
~(n + 8)y ÷6~fl — ~ (n + 2)T+
il’[~ (e + ~d)
411],
where Tare the [1] invariant corresponding to the l~and random vertex and tocharges the usual one,x = log ~2, ~is the correlation length. For the case Ed = 0 we have the point defect situation [1—3].Fig. 1 shows the phase plane for the system (3) (fig. la corresponds to e >0, Cd >0, fig. lb to = 0, ~d >0). The initial values of {T~l?’~are situated in the first quadrant of the phase plane near the coordinate origin. The random fixed point {Td, Ud} is a focus: 7d
=
ud
=
~ (~ + 3 ~d)I(~ — 1), ~ [(4— n)e + (n + 8)ed]/(n
(4) —
1).
The eigenvalues of eqs. (3) linearized in the vicinity of{Td,l~d}are
where Ccr is the concentration of lengthy defects (the 169
Volume 76A, number 2
PHYSICS LETTERS
a
\\\
4. When n = 1 the system (3) is accidentally degenerate. So it is necessary to take into account the next order of the Gell-Mann—Low functions and to carry
_________
Fig. 1. Phase
~
_________
plane for eqs.(3). (a) e >
0, Cd> 0; (b)
=
0,
Ed> 0. X ±j~= [16(n 1)]1[A A = —[3ne + (n + 8)~d]
±i(4B
—
B = 4(e When
+ 36d)(n
—
—
A 2)1/2] (5)
1)[(4
n) + (n + 8)d]
—
of~ie
When e = 0 we get the critical exponents for uniaxial magnets with lengthy defects. They are different from the mean field exponents. In conclusion, the author would like to note that a focus in the phase plane of the Gell-Mann—Low equations may be found in other systems too. Khmelnitskii [6] showed that the renormalization group equations for the phase transition at T = 0 in defect magnets possesses a stable focus fixed point. Korzhenevskii [8] found a focus-type fixed point while investigating the Potts model.
.
0
=
17 March 1980
(6) It is known that the presence of a stable focus in the phase plane leads to oscillations in the corrections to the scaling laws [6]. The susceptibility, for instance, is given by
The author is grateful to Yu A. Firsov, S.A. Ktitorov and B.N. Shalaev for useful discussions. References Khmelnitskii, Zh. Eksp. Teor. Fiz. 68(1975)1960. A.B. Harris and T.C. Lubensky, Phys. Rev. Lett. 33 (1974)
[1] D.E.
x~’
—
r1 exp [Ci-Xcos(v log r + p)],
(7)
where r = (T Tc)/Tc, T> T~C, p are non-universal constants. It can be shown that the , 6d expansions of the critical exponents are given by —
7
=
1
+
3
n
+
1 Sn
i~,~
4
+
—
~
+
“.‘
(8) a = [8(n
170
—
1)]~[(n —4)
—
(Sn +4)d]
+
...
.
[2]
1540. [3] T.C. Lubensky, Phys. Rev. Bil (1975) 3573. [4] A.P. Levanyuk, V.V. Osipov, A.S. Sigov and A.A.
Sobyanin, Zh. Eksp. Teor. Fiz.
76 (1979) 345.
[5] A.A. Abrikosov, L.P. Gor’kov and I.E. Dzyaloshinskii, Quantum field theory
methods in statistical physics (Benjamin, New York, 1963). [6] D.E. Khmelnitskii, Phys. Left. 67A (1978) 59. [7] B.N. Shalaev, Zh. Eksp. Teor. Fiz. 73 (1977) 2301. [8] A.L. Korzhenevskü, Zh. Eksp. Teor. Fiz. 75(1978)1474.